The paper classifies deformations of curves with inflections and vertices.
problem Deforming implicit curves with inflections and vertices.
method Classification of deformations for curves with 2 parameters.
result Generically, inflections and vertices emerge from singularities during deformations.
Abstract: Study of surface transitions and IDE inflections via contact geometry.
problem Understanding transitions on surfaces and implicit differential equations.
method Contact geometry and Legendrian properties of projections.
result List of unavoidable local phenomena on surfaces and IDE solutions.
We present an approach of computing the intersection curve C of two rational parametric surface §1(u,s) and §2(v,t), one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve G(v,t)=0. By analyzing the topology …
For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularit…
We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular, we use it to give a complete classification of biharmonic curves into real quadr…
The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
Study uses put-call parity to estimate cost of funding in equity derivatives markets.
problem Estimating the cost of funding in active equity derivative markets.
method Develops a method using European put and call prices to recover the implicit discount factor and cost of funding.
result Identifies the cost of funding in major equity markets, showing it is typically around 34 basis points above OIS.
Automatically explores geometric loci of curves using software networking.
problem Exploring hyperbolisms and geometric loci of plane curves.
method Parametric equations, Groebner bases, and elimination for deriving polynomial equations.
result Derives new constructions of lemniscates and other geometric loci.
A new method for optimizing non-decomposable metrics with constraints.
problem Optimizing complex machine learning objectives with thresholded constraints.
method Formulate rate-constrained optimization using the Implicit Function theorem and solve with gradient-based methods.
result Demonstrated effectiveness over existing methods on benchmark datasets.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame {T,V,U} along the curve, where T is the unit tangent vector field of the curve, U is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\ti…
Algorithm converts curves on ribbon surfaces to contact surgery diagrams.
problem Legendrian realization of curves on ribbon surfaces.
method Explicit algorithm to Legendrian realize homologically nontrivial curves.
result Any two Legendrian realizations of the same curve are Legendrian isotopic.
In this work we propose a differential geometric motivation for Nesterov's accelerated gradient method (AGM) for strongly-convex problems. By considering the optimization procedure as occurring on a Riemannian manifold with a natural structure, The AGM method can be seen as the proximal point method applied in this cur…
In the complex setting, let F(x,y,y′)=0 be an analytic or algebraic differential equation with y′-degree d. We deal with the qualitative study of such equations through the geometry of the planar d-web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discu…
Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Θ(n3/2) homotopy moves in the worst case. Our algorithm improves the best previou…
Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.
problem Constructing smooth C∗-actions on moduli spaces of super stable curves and maps of genus zero. method Using the implicit function theorem, proving smooth split atlases, and studying automorphism groups.
result Explicit descriptions of normal bundles to fixed loci in terms of spinor bundles and sections.
New surfaces found in 5D space.
problem Constructing smooth embedded special Legendrian surfaces in \(\mathbb S^5\).
method Combining implicit function theorem, loop algebra-valued meromorphic connections, and character variety analysis.
result First genus > 1 embedded special Legendrian surfaces in \(\mathbb S^5\).
We have discovered a "little" gap in our proof of the sharp conjecture that in Rn with volume and perimeter densities rm and rk, balls about the origin are uniquely isoperimetric if 0<m≤k−k/(n+k−1), that is, if they are stable (and m>0). The implicit unjustified assumption is that the g…
Given a Riemannian metric on the 2-sphere, sweep the 2-sphere out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show the following useful property (see Th…
This work evaluates deep generative models using RD curves, providing a more comprehensive quality assessment.
problem Quantitative evaluation of deep generative models is challenging, especially for implicit models.
method Proposes using rate distortion (RD) curves to evaluate and compare deep generative models, approximating the entire curve with similar computations to log-likelihood estimation.
result Approximating the entire RD curve provides a more comprehensive quality assessment than scalar-valued metrics.
This paper shows how to train only the implicit layer of overparameterized implicit neural networks.
problem Understanding how the implicit layer contributes to the training of overparameterized implicit neural networks.
method Restricting training to only the implicit layer and analyzing the generalization error for ReLU-activated networks.
result Global convergence is guaranteed even if only the implicit layer is trained, and gradient flow with proper random initialization can achieve small generalization errors.
Paper studies the theoretical equivalence between implicit and explicit neural networks in high dimensions.
problem Lack of theoretical analysis of implicit and explicit neural networks.
method Examined high-dimensional implicit neural networks and established their equivalence to explicit networks.
result Equivalence between implicit and explicit neural networks in high dimensions.
Gradient matching method estimates implicit regularization in complex deep learning systems.
problem Estimating implicit regularization in modern deep learning systems with complex modifications.
method Gradient matching methods to empirically estimate implicit regularization.
result Empirical estimation of implicit regularization in arbitrary networks, including dropout.
Continuous semi-implicit models enable faster training and better performance in generative modeling.
problem Slow convergence in hierarchical semi-implicit models during training.
method CoSIM, a continuous semi-implicit model that incorporates a continuous transition kernel for efficient training.
result CoSIM achieves superior performance on image generation tasks compared to existing methods.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
Study finds implicit government guarantee improves municipal investment bond ratings.
problem Questioning the objectivity of municipal investment bond ratings due to implicit government guarantee.
method Text mining of policy documents and PMC index model for implicit guarantee strength calculation.
result Implicit government guarantee boosts municipal investment bond ratings, especially in less developed regions.
Study shows SGD's generalization is not explained by implicit bias.
problem Explaining the generalization ability of overparameterized learning algorithms.
method Revisited Stochastic Convex Optimization with SGD, demonstrating limitations of implicit bias.
result No distribution-independent or distribution-dependent implicit regularizer can explain SGD's generalization.
This paper measures the intensity of implicit government guarantees using PMC index model.
problem Excessive local government debt due to implicit government guarantees.
method Text mining of policy documents related to municipal investment bonds, PMC index model.
result Recent policies have reduced the intensity of implicit government guarantees.
The paper explains implicit regularization in hierarchical tensor factorization and deep CNNs.
problem Understanding implicit regularization in complex neural network architectures.
method Theoretical analysis using dynamical systems to overcome challenges in hierarchy.
result Established implicit regularization towards low hierarchical tensor rank, equivalent to locality in CNNs.
In this paper, we describe the "implicit autoencoder" (IAE), a generative autoencoder in which both the generative path and the recognition path are parametrized by implicit distributions. We use two generative adversarial networks to define the reconstruction and the regularization cost functions of the implicit autoe…
Implicit discourse relation classification is of great challenge due to the lack of connectives as strong linguistic cues, which motivates the use of annotated implicit connectives to improve the recognition. We propose a feature imitation framework in which an implicit relation network is driven to learn from another …
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
Gradient descent on ReLU networks with square loss implicitly favors balanced weights.
problem Understanding implicit regularization in nonlinear neural networks with regression losses.
method Analyzing gradient descent dynamics on ReLU networks with square loss.
result It is impossible to characterize the implicit regularization of ReLU networks with square loss by any explicit function of model parameters.
New insights into how deep models generalize, focusing on matrix factorization.
problem Understanding how deep models generalize and why they work well.
method Using Morse functions and dynamical systems to study implicit regularization.
result Solved a conjecture on implicit regularization in matrix factorization.
We extend the existing framework of semi-implicit variational inference (SIVI) and introduce doubly semi-implicit variational inference (DSIVI), a way to perform variational inference and learning when both the approximate posterior and the prior distribution are semi-implicit. In other words, DSIVI performs inference …
Deep equilibrium models converge globally without explicit computation.
problem Global convergence of deep learning models with implicit layers.
method Analysis of gradient dynamics and proof of convergence rate.
result Deep equilibrium models converge to global optimum at a linear rate.
Proposes efficient, modular method for implicit differentiation.
problem Implicit differentiation of optimization problems.
method Automatic implicit differentiation using autodiff and implicit function theorem.
result Automatic differentiation of optimization problems is made easier and more modular.
Sayer uses implicit feedback to optimize system policies.
problem Leveraging implicit feedback to improve system policies is difficult due to bias and incompleteness.
method Sayer combines randomized exploration and unbiased counterfactual estimators to evaluate and train new policies using implicit feedback.
result Sayer can accurately evaluate and train new policies that outperform existing ones.
This paper analyzes implicit bias in Deep Linear Discriminant Analysis.
problem The implicit bias of Deep Linear Discriminant Analysis.
method Analyzing gradient flow on a L-layer diagonal linear network.
result Under balanced initialization, the network transforms additive updates into multiplicative updates, conserving the (2/L) quasi-norm.
The paper analyzes implicit regularization in tensor factorization using neural networks.
problem Understanding implicit regularization in tensor factorization.
method Dynamical systems perspective and gradient descent analysis.
result Gradient descent induces a form of greedy low tensor rank search.
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
Recent progress in variational inference has paid much attention to the flexibility of variational posteriors. One promising direction is to use implicit distributions, i.e., distributions without tractable densities as the variational posterior. However, existing methods on implicit posteriors still face challenges of…
Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent stru…
IDAC improves reinforcement learning efficiency by modeling implicit distributions.
problem Improving sample efficiency in reinforcement learning algorithms.
method IDAC uses two DGNs for a distributional critic and a semi-implicit actor to model implicit policy distributions.
result IDAC outperforms state-of-the-art algorithms on OpenAI Gym environments.
Global implicit function theorem for Fréchet spaces, solving derivative loss problems.
problem Solving initial value problems with derivative loss in Fréchet spaces.
method Global implicit function theorems for Keller's Cc1-mappings in Fréchet spaces, applied through submersions and transversality. result Global existence and uniqueness of solutions to initial value problems with derivative loss.
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion …
New proof shows norms can't explain deep learning's implicit regularization.
problem Understanding the implicit regularization in deep learning.
method Mathematical proof on matrix factorization problems.
result Implicit regularization drives norms towards infinity, suggesting rank minimization is key.